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9    MEASURABLE FUNCTIONS       149

(c)    Take X to be the circle U : z\ = 1 in C, and ju, to be the image of Lebesgue
measure under the mapping t\-+e2nit of [0, 1] onto U. If 6 is irrational, show that the
mapping u : z\-+e2ntez leaves ft invariant and is ergodic with respect to yi. (Use BohPs
theorem, Section 13.4, Problem 7.)

(d)    Suppose that X is compact and u ergodic with respect to jit. If A and B are
jLt-measurable subsets of X, show that

lim - I]' p,(u~k(A) n B) = jLc(A)/x(B)/ju(X).

n-*oo n k= 0

Conversely, if this formula is satisfied for all pairs of measurable subsets A, B in X,
then u is ergodic with respect to p.

(e)    Suppose that X is compact and u ergodic with respect to /x. If a measurable
function/^ 0 is such that

lim -x;1 /<>*(*))

n-*oo  ft fc = 0

exists almost everywhere, then /is integrable. (Consider /as the limit of an increasing
sequence of bounded functions.)

(f)    Suppose that X is compact, that u is ergodic with respect to /x, and that
Supp(ju) = X. Show that, for almost all # <= X, the set of points if(x) (n ^> 0) is dense
inX.

14. Let X be a compact space, u : X -> X a continuous mapping, and p, a positive measure
on X, invariant with respect to u and of total mass 1 . Let A be a jLt-measurable subset
of X. For each x e A, let n(x) denote the smallest integer n J> 1 such that un(x) e A.
If we put An = u~n(A) and Bn = w ~"(X  A), then the set of points x e A for which
n(x)  m is A0 n Bj n B2 r\  -  n Bm_] n Am, and hence the function x\*n(x) is
measurable.

(a)   Put a0 = 1 and aw = /x(B0 n BI n    n Bm.~i) (m ^> 1). Show that
BI n B2 n    n Bm_i n Bm n Am + i) = am  2am+1 -j-

for all m ;> 0.

(b) Show that the series whose general term is am  am+1 is convergent and that its
terms form a decreasing sequence (interpret ocm  am+i as the measure of a set).
Deduce that lim m(ocm  am + 1) == 0.

(c)   Let Bo, = fl B"  Snow tnat

' n(x)

C'Kac's theorem": use (a) and (b)). Consider the case where u is ergodic with respect
to ju, and /x(A) > 0.

(d)   Suppose that u is bijective and ergodic with respect to /x, and that jtt(A) > 0.
Let Em be the set of points x e A such that n(x) = m. Show that the complement in

A of  0 Em is negligible (cf. Problem 11 (a)). Show that the sets up(Em) for m ^ 1

m- 1

and 0 <; p < m are mutually disjoint, and that the complement of their union in X
is negligible ("Kakutani's skyscraper")-consider first the case where/is bounded; then pass to the general case by observing to show that it implies that there exists a rational number r